An algebra of pseudodifferential operators
The author considers pseudo-differential operators in \(\mathbb{R}^n\) with symbol \(a(z)\), \(z= (x, \xi)\), satisfying the following property: \[ \sup_j |(f_j a)\hat{}(\zeta)|\in L^1(\mathbb{R}^{2n}_\zeta), \] where \(j\) runs over the lattice of the \(2n\)-tuples of integers, and \[ f_j(z)= f_0(z- j)\in C^\infty_0(\mathbb{R}^{2n}) \] with \(\sum_j f_j= 1\). Such symbols turn out to be bounded continuous functions in \(\mathbb{R}^{2n}\). The author proves that the corresponding class of pseudo-differential operators: 1) is independent of the \(t\)-quantization, \(t\in \mathbb{R}\), \[ \text{Op}_t(a) u(x)= (2\pi)^{- n} \int e^{i(x- y)\xi} a(tx+ (1- t) y,\xi) u(y) dy d\xi; \] 2) is stable under composition; 3) consists of \(L^2\)-bounded operators.
scientific article; zbMATH DE number 817686